Removal of multiples from seismic reflection data is an important preprocessing step before conventional seismic imaging and inversion in most onshore and offshore environments. While many methods have been developed and successfully used to remove free-surface multiples, internal multiple attenuation remains a challenge when working in land and complex marine environments. The inverse scattering series (ISS) internal multiple algorithm is a data-driven tool to predict all orders of internal multiples for all horizons at once, without requiring subsurface information. However, use of the multidimensional version of this algorithm has been limited due to high computational cost, which increases with the maximum output frequency in the prediction. Even with the recent advances in computer hardware, the cost of the multidimensional algorithm remains expensive. To overcome this problem, we use the quasi-Monte Carlo integration technique that can significantly improve the computational efficiency of the multidimensional ISS internal multiple algorithm. The efficiency is improved by reducing the number of samples being evaluated and combining multiple integrals into a single summation.
Conventional seismic imaging and parameter estimation (inversion) assume that the data contain only primaries, which are singly reflected events that propagated through the subsurface. Multiple reflected events in the recorded wavefield are considered noise because they do not satisfy the conventional assumptions of imaging and inversion, can have destructive interference with what is considered signal (primaries), create false images and/or be misinterpreted as primaries. Hence, the need for demultiple algorithms. Multiples are a type of multiple reflected events with at least one downward reflection, and are distinguished from ghost events because multiples propagate down from the source and are recorded as upgoing waves at the receivers. Depending on the location of the downward reflections, multiples are divided into free-surface and internal multiples. Free-surface multiples have at least one downward reflection at the free surface. Internal multiples have all downward reflections below the measurement surface (in surface seismic acquisitions). When exploring offshore areas, the most dominant multiples are associated with free-surface reflections; thus performing free-surface demultiple is often sufficient. Moreover, in onshore seismic exploration, the internal multiples dominate and are often an impediment to obtaining a reliable image and interpretation of the subsurface. Free-surface demultiple technology has reached a mature state in seismic processing; there is a toolbox available with effective effective, practical algorithms (see, e.g., Verschuur et al., 1992; Carvalho et al., 1992; Dragoset and Jericevic, 1998; Liu and Sacchi, 2002; Weglein and Dragoset, 2005). On the other hand, effective and efficient algorithms for internal multiple removal still need development to be robust and efficient in practice. The ISS internal multiple attenuation algorithm was first proposed by Araújo et al. (1994) and Weglein et al. (1997). It is a data-driven algorithm, independent of subsurface information, that predicts internal multiples for all horizons at once. This algorithm predicts the correct traveltimes and an approximated amplitude of the true internal multiples in the data. RamÍrez and Weglein (2005) extended the theory from attenuation towards elimination by improving the amplitude prediction.